Local Geometry of Real and Complex Analytic Mappings

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The proposed research program is concerned with the local geometry of real and complex analytic mappings and their images. It lies at the interface of analytic geometry, commutative algebra, singularity theory and theory of functions of several complex variables, and involves problems in all these directions. The program builds on very successful work done under my present NSERC discovery grant.The two main objectives of the program are:(a) Establishing of a computable classification of singularities of analytic and polynomial mappings.(b) Development of a singular CR geometry.The first of the above long term goals is concerned with the local geometry of mappings from the point of view of singularity theory.Our main idea is to reduce the study of the geometric complexity of a map to an automated calculation of certain algebraic invariants. To achieve this goal, we need to develop new criteria that would characterize the geometry of a map in terms of algebraic properties of certain modules associated with the map, which can be verified by means of computer algebra methods. Examples of such criteria are our recent characterizations of openness and flatness of maps. Our approach is to study degeneracies (or discontinuities) in the family of fibres of a given map. These are often too subtle to be detected on the algebraic level, and so we need to amplify these discontinuities to the extent that they get reflected in algebraic properties of certain modules associated with the map. This can be done, for example, by passing to fibred powers of the map.The study of local invariants classifying the mapping singularities is a well established and active area of research. The novelty of our approach lies in its effectiveness, that is, the emphasis on computability. Our most recent results on finite determinacy of flatness and other local properties prove that this approach may be successful even in the transcendental (i.e., non-polynomial) case.Our second long term goal concerns the application of local analytic geometry to the study of real structures in complex ambient spaces. In the non-singular setting, this area of study is known as CR geometry, which can be viewed as a branch of the classical analysis of functions in several complex variables. In this proposal, we consider singular real analytic (even semianalytic) objects in complex spaces. The importance of this approach lies in the fact that such singular sets appear naturally in complex analytic considerations (e.g., as boundaries of complex domains).Singular analytic sets, of course, are not CR manifolds themselves. However, as we showed recently, they admit a stratification into CR manifolds enjoying some very nice differential and algebro-geometric properties. This discovery forms a firm basis for the development of CR geometry in the singular context, and allows us to use the methods of semialgebraic and semianalytic geometry.